In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by convergence of measures, consider a sequence of measures μn on a space, sharing a common collection of measurable sets. Such a sequence might represent an attempt to construct 'better and better' approximations to a desired measure μ that is difficult to obtain directly. The meaning of 'better and better' is subject to all the usual caveats for taking limits; for any error tolerance ε > 0 we require there be N sufficiently large for n ≥ N to ensure the 'difference' between μn and μ is smaller than ε. Various notions of convergence specify precisely what the word 'difference' should mean in that description; these notions are not equivalent to one another, and vary in strength.
Three of the most common notions of convergence are described below.
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Convergence in measure is either of two distinct mathematical concepts both of which generalize the concept ofconvergence in probability. Let f , f n...
weak convergence may refer to: Weak convergenceof random variables of a probability distribution Weak convergenceofmeasures, of a sequence of probability...
Look up convergence, converges, or converging in Wiktionary, the free dictionary. Convergence may refer to: Convergence (book series), edited by Ruth...
notions ofconvergenceof sequences of random variables, including convergence in probability, convergence in distribution, and almost sure convergence. The...
pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to...
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergenceof a sequence of functions...
Convergenceof Probability Measures is a graduate textbook in the field of mathematical probability theory. It was written by Patrick Billingsley and published...
|} is the total variation measureof μ {\displaystyle \mu } . Very often, the measures in question are probability measures, so the last part can be written...
false in general. Convergenceofmeasures in the Radon metric is sometimes known as strong convergence, as contrasted with weak convergence. Radonifying function...
better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of the convergence in Lp in terms ofconvergence in measure and a condition...
mathematical field of analysis, uniform convergence is a mode ofconvergenceof functions stronger than pointwise convergence. A sequence of functions ( f...
made extensive use of weak convergence. The early pioneers of functional analysis did not elevate norm convergence above weak convergence and oftentimes viewed...
flexibility. For instance, there are various natural notions ofconvergenceofmeasures, and it is not immediately clear whether they arise from a topological...
In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a convergence that satisfies certain...
defined. Convergence implies "Cauchy-convergence", and Cauchy-convergence, together with the existence of a convergent subsequence implies convergence. The...
{\displaystyle L_{1}} which is central in dominated convergence. Several textbooks on real analysis and measure theory use the following definition: Definition...
numbers. That is, Fn converges to F pointwise. Glivenko and Cantelli strengthened this result by proving uniform convergenceof Fn to F by the Glivenko–Cantelli...
variate Controlling for a variable ConvergenceofmeasuresConvergenceof random variables Convex hull Convolution of probability distributions Convolution...
Process-based convergence is when the convergence is due to similar forces of natural selection. Earlier methods for measuringconvergence incorporate ratios of phenotypic...
a branch of pure mathematics. Convergence is not necessarily given in the general case, and certain criteria must be met for convergence to occur. Determination...
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